DE102023128976A1 - Method for estimating the unbalance distribution of a rotor - Google Patents

Abstract

In a method for estimating the unbalance distribution in reference planes of a rotor, one or more causes of unbalance are represented for each component of the rotor with their respective unbalance distributions in the reference planes of the rotor. The unbalances of each component are then taken into account with a selectable quantile limit in an axial position and an assembly of a large number of rotors with its components is simulated. A Monte Carlo simulation of the large number of rotors with their various distributions of causes of unbalance can be carried out from this and the statistical unbalance distribution of the large number of rotors in their reference or compensation planes can be derived.

Description

The invention relates to a method for estimating the unbalance distribution in reference planes of a rotor, in which each component of the rotor is assigned one or more causes of unbalance with their respective unbalance distributions in the reference planes of the rotor.

After the very demanding part of rotor modeling, especially for engine rotors, another essential design step must be carried out by deriving the production drawings. Each individual part of a rotor requires a precise definition of its reference surfaces, to which all other form and shape tolerance limits can be related. Roundness, concentricity, coaxiality, radial and axial runout describe the permissible misalignment of all rotor parts. If these effects are analyzed from the point of view of balancing, most causes of imbalance can be assigned to individual geometric tolerance limits.

Balancing always begins with the correct definition of the axis of rotation of a rotor and a rotor coordinate system that defines the phase of any location on the rotor circumference.
Based on this definition, three types of imbalances can be analyzed:

Intrinsic imbalance

For each individual component, a maximum possible initial unbalance is calculated/estimated, based on the manufacturing process, manufacturing tolerances and the complexity of the component. If a balancing tolerance is specified on the drawing, this value is used: U = m * r

Unbalance due to concentricity

If a component is mounted on a rotor on a surface that has a concentricity, an imbalance occurs: U = e * m

Unbalance due to clearance

If a component with a clearance fit is mounted on a rotor, an imbalance also occurs: U = e * m

There are other reasons, e.g. inclinations of rotor components due to axial runout of disk-like components, which lead to moment imbalances.

In individual cases, there may be other causes of unbalance, such as component masses, eccentricity, concentricity, coaxiality and radii, and each of these dimensions is associated with a form and shape tolerance limit. Using the error propagation law, the specific uncertainties of a form and position tolerance causing unbalance can be summarized to the expected distribution of the unbalance at the end of the manufacturing process.

The unbalance is a vector described as polar coordinates with a magnitude and a phase. The phase is related to the rotor coordinate system and in many cases uses a specific rotor characteristic. In addition, each unbalance vector is associated with a specific axial position.

Each sum of different causes must be related to specific rotor planes. This requirement requires a plane transfer by applying the lever law. Each cause of unbalance must be transferred to the reference planes of the rotor to summarize.

The object of the invention is to provide a method which makes it possible to estimate the imbalance of a rotor during production only on the basis of production drawings and to use this information for any necessary unbalance compensation.

The problem is solved by the teaching of claim 1. Preferred embodiments of the teachings can be found in the dependent claims.

The invention relates to a method for estimating the unbalance distribution in reference planes of a rotor, in which each component of the rotor is assigned one or more causes of unbalance with their respective unbalance distributions in the reference planes of the rotor,
Unbalances of each component are taken into account as a selectable quantile, in particular a selectable quantile limit, advantageously the 99% quantile, in an axial position,
an assembly of a large number of rotors is simulated using Monte Carlo simulation with its components,
the Monte Carlo simulation is carried out with a large number of rotors with their different distributions of unbalance causes in the production process and from this the statistical unbalance distribution of the large number of rotors in their reference planes is derived, whereby in particular the statistical unbalance distribution in both reference planes is derived from the large number of rotors.

When developing a rotor, it is important to design a rotor with an expected unbalance distribution in order to be able to plan for unbalance compensation in production. The difficulties arise when designing components using CAD software, which does not represent the presence of unbalance or the ability to achieve a balanced state for each rotor design. The manufacturing processes ensure that the series production steps are within the tolerance limits specified in the production drawings. The method according to the invention can ensure that the unbalance contributions estimated in the design phase can be compensated for by unbalance compensation in later production.

The flawless operation of a rigid rotor, i.e. with low vibration and unbalance, depends on effective compliance with the concentricity and eccentricity of various rotor components. The determination of these shape and position tolerances must be taken into account during design to ensure that the rotors to be manufactured ultimately remain within the permissible residual unbalance range.

Finite element methods can be used to estimate the effects of different unbalance distributions along the rotor axis and their excitation of the elastic behavior of the rotor shaft. Predictive analyses with probabilities determine the contributions to the vibration levels of the assembly. A method is proposed with which the effects of permissible manufacturing uncertainties in the earliest phases of rotor design on the subsequent assembly processes can be taken into account.

In a preferred design, reference planes are introduced as compensation planes, so that the unbalance distribution must be designed to be correctable by a balancing step.

It is intended that the causes of unbalance, unbalance values and axial positions of each component of the rotor are determined.

Furthermore, it is intended that imbalances of the components are related to bearing levels.

With the help of the Monte Carlo simulation, it is also advantageously possible to simulate a two-dimensional distribution of a systematic and a random unbalance deviation of all production steps, whereby the definition of shape and bearing tolerances of the components makes it possible to keep the final unbalance state within the desired limits.

It is advantageous that an intrinsic unbalance of each component, especially important ones, is determined.

Furthermore, it can be advantageous that each individual geometric tolerance limit that defines an unbalance amount is represented with an evenly distributed phase position across all rotors in the simulation.

Interpretation of form and position tolerance limits

Each geometric tolerance limit represents a specific quantile of the corresponding probability distribution. Different types of distributions can be considered.

In general, any quantile can be used, but for unmonitored geometric tolerances, the value 99% as a quantile fits well with experience.

The above-mentioned causes of unbalance cause a circular uncertainty range in a polar diagram with the amount of unbalance as a ring-shaped boundary. But the vector characteristics, in particular the individual phase information, are still missing. This problem is solved by assuming a rectangular distribution of the phase position for each individual cause of unbalance.

Combination of unbalance distributions

By using a Monte Carlo simulation, different causes of imbalance, including different distributions, can be summed up in the reference planes of the rotor. The causes of imbalance can be of a systematic or random nature. This also allows balancing processes in the production process to be simulated.

A typical form for unbalance distributions is the two-dimensional normal distribution. Furthermore, shape and position tolerances can be represented, for example, by uniform, normal or Weibull distributions. The summation of the distributions of various unbalance contributions will also form a distribution, which is ideally shown in a probability diagram. From this, the respective achievable unbalance limits can then be derived in selectable quantiles.

Application of Monte Carlo simulation

The quantile limits of unbalance contributions cannot be transformed into other levels. The lever law for converting unbalances into other levels requires vectors for the level transformations.

The solution to this problem is the use of Monte Carlo simulation. For each unbalance distribution of each cause of unbalance, a selectable number of elements of this distribution form are calculated. These individual elements are vectors that can be converted to other levels, unlike quantile limits.

The method advantageously simulates a virtual series production process of the rotor. Starting with the virtual assembly of, for example, 10,000 rotors, a representative result of the final unbalance distribution in the reference planes of the rotor is already obtained. 10,000 components are also required for this assembly process. Each component is assigned one or more causes of unbalance and their distributions. This makes it possible to estimate the expected unbalance distribution in the reference planes of the rotor. If these reference planes are introduced in a design as compensation planes, it is possible to see whether the unbalance distribution can be corrected by the final balancing step. The result after the virtual compensation step can also be evaluated in the reference planes - usually the bearing planes.

• 1 schematische Ansicht eines Rotors,
• 2 Schnittdarstellung des Rotors gemäß 1,
• 3 eine schematische Darstellung der Welle,
• 4 eine schematische Darstellung der Scheibe auf der Welle,
• 5 eine Schnittdarstellung des Rotors gemäß 1 mit den Bereichen A, B,
• 6 Darstellung der Wahrscheinlichkeit in Relation zur Unwucht,

The invention is explained in more detail below using an embodiment of the invention, which is shown in the drawing.

• 1 schematic view of a rotor,
• 2 Sectional view of the rotor according to 1 ,
• 3 a schematic representation of the wave,
• 4 a schematic representation of the disc on the shaft,
• 5 a sectional view of the rotor according to 1 with the areas A, B,
• 6 Representation of the probability in relation to the imbalance,

Figure 1: Introduction of a rotor example

The following example of a rotor is purely fictitious, is not intended to represent a real assembly and any similarities are purely coincidental. The geometric dimensions and tolerances marked are only an excerpt and do not describe all rotational relationships or possible causes of imbalances that may occur. The rotor unit consists of four main parts that are rotationally symmetrical: shaft 1, disc 2, cone 3, hollow ring 4.

The main shaft 1 has two bearing seats and forms the axis of rotation. The disk 2 sits directly on the shaft 1 and is followed by the cone 3. The hollow ring 4 is positioned by the cone 3 and fastened with screws. The axial positions of the component-specific centers of gravity must be precisely defined, as this is where the planes of the causes of imbalance are located. The combination of all uncertainties is represented in the bearing planes. 2 and the following table describe the overview of the plane and center of gravity distances. Description COG - distance in mm Mass in kg Wave 274 8th disc 115 22 cone 261 11 Bright Ring 385 11.2 Screw (1x) 327 0.025 Left camp 50 / Right camp 500 / Assembly 230 52.68

Explanation of the causes of imbalance

Various causes of imbalance are listed below. For the sake of clarity, only a few representative causes of imbalance are considered.

The imbalance influence of the shaft

• • Die Welle 1 selbst, aber aufgrund ihrer präzisen Bearbeitung kann dieser Betrag vernachlässigt werden.
• • Der Sitz der Scheibe 2.
• • Der Sitz des Konus 3.

Shaft 1 contributes to the imbalance in three ways:

• • The shaft 1 itself, but due to its precise machining this amount can be neglected.
• • The seat of the disc 2.
• • The seat of the cone 3.

A typical reason for static imbalance is runout. This results in an eccentricity for the mass of the assembled component. Although these runouts can be minimized, they cannot be eliminated for cost, production time and technical reasons.

$$U_{disc}=0.005mm\cdot \mathrm{22,000}g=110gmm$$

The shaft seats for disc and cone are each tolerated with run=1 0 µm concentricity error and result in an eccentricity e = run/2 = 0.005 mm (see 3 ). For both parts A, B of shaft 1, the static unbalance is calculated with $$U=e\cdot m$$

$$U_{disc}=0.005mm\cdot \mathrm{22,000}G=110Gmm$$

The cone 3 also holds the hollow ring 4. $$U_{cOne}=0.005mm\cdot \left(\mathrm{11,000}G+\mathrm{11,200}G\right)=111Gmm$$

The unbalance influence of disc 3

• • Die Eigenunwucht der Scheibe 3 ist statisch ausgewuchtet mit einer zulässigen Toleranzgrenze von 35 gmm.
• • Der laterale Run-out der Scheibe 3.
• • Die Scheibe 3 bringt durch den radialen Run-out der Wellenpassung eine Unwucht mit sich. Diese Unwucht wird auch durch das Gewicht des Hohlrings 4 verursacht. Dieser Beitrag wurde im Absatz für die Welle 1 berücksichtigt.

Disc 3 contributes to the imbalance in three ways:

• • The inherent unbalance of disc 3 is statically balanced with a permissible tolerance limit of 35 gmm.
• • The lateral run-out of the disc 3.
• • The disc 3 introduces an imbalance due to the radial run-out of the shaft fit. This imbalance is also caused by the weight of the hollow ring 4. This contribution has been taken into account in the paragraph for shaft 1.

The fit for the lateral runout error of disk 3 does not cause a static imbalance, unlike radial errors. The center of gravity remains on the axis of rotation. An inclined position leads to an unbalance moment. This unbalance moment is independent of its axial position and can be freely shifted to the axis of rotation. The resulting moment unbalance is divided by the plane distance considered, e.g. the distance between the compensation or bearing planes, and thus gives the contribution of an unbalance pair in the reference planes.

The axial runout error of disk 3 in the assembled state is tolerated at 0.01 mm on the outer diameter. It describes the misalignment of disk 3 in relation to the bearing seats and leads to an unbalance moment or an unbalance pair in the reference planes.

The calculation requires the mass moments of inertia around the center of gravity and the x and z axes.

$$U_c=\frac{0.01mm}{350mm\cdot 450mm}\cdot \left|\left(\left(\mathrm{173,9}-\mathrm{344,6}\right)\cdot {10}^6\cdot gm{m}^2\right)\right|$$

$$U_c=10.84gmm$$

The moment unbalance Uc is calculated (see 4 ): $$U_c=\frac{H}{d\cdot a}\cdot \left|\left(I_x-I_z\right)\right|$$

$$U_c=\frac{0.01mm}{350mm\cdot 450mm}\cdot \left|\left(\left(173.9-344.6\right)\cdot {10}^6\cdot Gm{m}^2\right)\right|$$

$$U_c=10.84Gmm$$

The unbalance moment U _m (independent of reference planes) is defined as $$U_M=U_c\cdot a=10.84Gmm\cdot 450mm=4875.24Gm{m}^2$$

The unbalance influence of cone 3

• • Die Eigenunwucht des Konus 3 ist statisch ausgewuchtet mit einer zulässigen Toleranzgrenze von 40 gmm.
• • Der radiale Rundlauffehler des Konus 3 durch den radialen Rundlauffehlers der Wellenaufnahme. Diese Unwuchtursache umfasst auch das Gewicht des Hohlrings 4. Dieser Beitrag wurde für die Welle 1 berücksichtigt.

Cone 3 contributes to the imbalance in two ways:

• • The inherent unbalance of cone 3 is statically balanced with a permissible tolerance limit of 40 gmm.
• • The radial runout of cone 3 due to the radial runout of the shaft mount. This cause of imbalance also includes the weight of the hollow ring 4. This contribution was taken into account for shaft 1.

The unbalance influence of the hollow ring 4

• • Die Montage des Hohlrings 4 auf dem Konus 3.
• • Die Unwucht des inneren Bereichs durch eine exzentrische Innenfläche,
• • Der Hohlring 4 verursacht eine Unwucht durch die Wellenbefestigung des Konus 3, dieser Betrag ist im Absatz der Welle 1 enthalten, andere mögliche Effekte werden in diesem Beispiel vernachlässigt.

The hollow ring 4 contributes to the imbalance in three ways:

• • The assembly of the hollow ring 4 on the cone 3.
• • The imbalance of the inner area due to an eccentric inner surface,
• • The hollow ring 4 causes an imbalance due to the shaft fastening of the cone 3, this amount is included in the shoulder of the shaft 1, other possible effects are neglected in this example.

For technical and handling reasons, parts are often assembled with a clearance fit. This results in a degree of freedom with regard to the off-center positioning of the center of gravity for the attachment part.

$$U_{hollowring}=0.029mm\cdot \mathrm{11,200}g=324.8gmm$$

The fit between the cone 3 and the hollow ring 4 is a clearance fit with an h6/H6 combination. With a nominal diameter of 210 mm, a maximum clearance of 58 µm is possible, or an eccentricity of 29 µm. The static unbalance is calculated using the mass of the respective component: $$U=e\cdot m$$

$$U_{HOllOwrinG}=0.029mm\cdot \mathrm{11,200}G=324.8Gmm$$

Runout errors of the inner and outer surfaces of hollow bodies often lead to large unbalances, as they are very often underestimated. The static unbalance is the product of the eccentricity of the inner volume, the. This approach assumes that the outer diameter is related to the surfaces A and B of the shaft (see 5 ).

$$U_{Hollow}=\frac{\left(D_a{}^2-D_i{}^2\right)}{4}\cdot h\cdot \pi \cdot \rho \cdot \frac{run}{2}$$

$$U_{Hollow}=\frac{{\left(260mm\right)}^2}{4}\cdot 140mm\cdot \pi \cdot 0.0078\frac{g}{m{m}^3}\cdot \frac{0.02mm}{2}$$

$$U_{Hollow}=579.77gmm$$

If the inner contour has a runout error of 0.02 mm to the reference surfaces A and B, the possible unbalance is calculated as follows: $$D_a=0$$

$$U_{HOllOw}=\frac{\left(D_a{}^2-D_i{}^2\right)}{4}\cdot H\cdot \pi \cdot \rho \cdot \frac{ru\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102023128976A1\_0023.png,DE102023128976A1\_0024.png"\; state="\{\{state\}\}">n</figure-callout>}}{2}$$

$$U_{HOllOw}=\frac{{\left(260mm\right)}^2}{4}\cdot 140mm\cdot \pi \cdot 0.0078\frac{G}{m{m}^3}\cdot \frac{0.02m\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="DE102023128976A1\_0023.png,DE102023128976A1\_0024.png"\; state="\{\{state\}\}">m</figure-callout>}}{2}$$

$$U_{HOllOw}=579.77Gmm$$

The imbalance influence of the screws

Symmetrical arrangements can also cause imbalances because their positions or masses also form a distribution.

The hollow ring 4 is attached to the cone 3 with 16 screws. One screw has a nominal mass of 25 g and a mass distribution of ± 0.5 g. This value is assumed to be the 99% quantile (2.576s). The standard deviation is therefore s _m = Δm/2.576 = 0.194 g. The statistical sum of all 16 screws gives the standard deviation of the components by: $$s_{cOmp}=\frac{\sqrt{16}}{\sqrt{2}}$$

The 99% limit of unbalance for the 16 screws is calculated as follows: $$U_{99\%\mathrm{,16}Screws}=\frac{\sqrt{16}}{\sqrt{2}}\cdot 3.04\approx 8.6$$

$$d{U}_{16Screws}=8.6\cdot 0.194g\cdot 118mm$$

$$d{U}_{16Screws}=196.87gmm$$

With 16 screws, the total unbalance contribution (limit value) is: $$d{U}_{16Screws}=U_{99\%\mathrm{,16}Screws}\cdot s_m\cdot r_{eeeectiveradius}$$

$$d{U}_{16Screws}=8.6\cdot 0.194G\cdot 118mm$$

$$d{U}_{16Screws}=196.87Gmm$$

The unbalance influence of the bearings

• • Parallelverschiebung $\overrightarrow{e}$
  
  der Drehachse → statische Unwucht
• • Neigung $\overrightarrow{\alpha }$
  
  (Neigung) der Rotorachse um den Schwerpunkt → Unwuchtmoment

The majority of imbalances in ball bearings are caused by the eccentricity of the inner ring. Different quality levels of ball bearings and their diameters define permissible eccentricities of the inner rings:

• • Parallel shift $\overrightarrow{e}$
  
  the rotation axis → static unbalance
• • Tilt $\overrightarrow{\alpha }$
  
  (Inclination) of the rotor axis around the center of gravity → unbalance moment

und Neigung $\overrightarrow{\alpha }$

sind in ihrer reinen Form sehr unwahrscheinlich. Es ist offensichtlich, dass sich diese Fälle überlagern und in der Realität eine Mischung aus beiden vorkommt.These two special cases parallel shift $\overrightarrow{e}$

and inclination $\overrightarrow{\alpha }$

are very unlikely in their pure form. It is obvious that these cases overlap and in reality a mixture of both occurs.

These deviations of the inner bearing rings can lead to an unknown mixture of parallel displacement and inclination of the axis of rotation of the entire rotating group. In this approach, the influence of the two ball bearings must be considered statistically independent of each other. With the parameters of the axial position of the two ball bearings, the mass of the rotor, the polar and equatorial moment of inertia, the unbalance caused by the eccentricity of the inner rings of the ball bearings can be derived.

Combination of all unbalance contributions

After determining the causes of the imbalance, their imbalance values and axis positions, all influences are summarized in a simulation. Each cause of imbalance is identified and a virtual assembly process for a large number of rotors, e.g. 10,000 pieces, can be started. Each component contains the imbalances assigned above as a 99% quantile in a specific axial position. A virtual assembly process for all components, e.g. 10,000 rotors, ultimately leads to a sum of all rotor components. Description COG - distance in mm Unbalance [gmm] Moment [gmm ² ] Wave Wave intrinsic - 0 Seat of the disc 115 110gmm Shaft cone + hollow ring 323 111 gmm disc Disc intrinsic 115 35gmm Lateral concentricity - 4875 ^gmm2 cone Cone intrinsic 261 40gmm Hollow ring Adaptation to the cone 385 325gmm Hollow space 385 580gmm Screw (16x) 327 197 gmm Bearings , inner rings 50 & 500 Concentricity 10 µm

The unbalances of all components are related to the bearing levels, which are referred to as level A for the left bearing and as level B for the right bearing (see 6 ).

The 99% quantile for level A is: 246gmm,

The 99% quantile for level B is: 526gmm,

In the case of simulating a complete rotor manufacturing process including possible balancing processes, the expected unbalance distributions in the bearing planes are available at the end of rotor manufacturing. Confirmation of the capability of an unbalance compensation process must be provided by specifying the unbalance distribution in the compensation planes. In order to estimate the behavior of shaft-elastic rotors, it may be necessary to calculate only rotor assemblies along the shaft length in order to be able to use their unbalance contribution for the respective shaft section in a rotordynamic calculation.

Pareto diagram

In addition to the information about the unbalance distribution in axial reference planes, this calculation of a so-called unbalance budget also produces a list of causes of unbalance, which, sorted by their unbalance contribution, identify the strongest driver of the rotor unbalance. If the causes of unbalance are sorted by their contribution in the present example, it becomes clear that the concentricity of the inner contour of the hollow ring is a main cause of the unbalance in plane B. The inner contour is not a functional surface and therefore roughly tolerated. However, due to the imbalance caused by the rough toleration of the inner contour, a tolerance restriction may be necessary without the function of the inner contour requiring this.

Extension to elastic rotors with shaft

The above-mentioned plane transformation into reference planes by applying the lever law assumes that the rotor behaves like a rigid body. This situation is a special case and not a general rotor condition. If all the forces that occur due to the causes of imbalance are not strong enough to bend the rotor in the specified speed range, the rotor is to be considered rigid.

If a rotor is operated in a speed range which, for the calculated unbalances in the speed range, leads to forces that bend the shaft, the axial distribution of the unbalance causes in relation to the eigenmodes becomes important.

The application of the unbalance budgeting method described here enables the unbalance prediction of rotor assemblies in specific axial positions for the calculation of the excitation for rotor bending by finite element methods.

The unbalance of rotors is mainly determined by the geometric shape and the shape tolerances established for the manufacturing process. The CAD model does not indicate any random unbalance condition. But in this design phase of rotors, the balancing process should be defined, unfortunately without knowing what amount of unbalance needs to be compensated.

With the method according to the invention, it is now possible to derive the unbalance distribution of the subsequent manufacturing process in advance. In addition, no prototypes are required for initial unbalance determinations, which are usually manufactured using other processing methods in a series process. The sequence of the manufacturing processes in the rotor structure can also be checked if the production drawings are prepared accordingly.

Claims (7)

Method for estimating the unbalance distribution in reference planes of a rotor, in which each component of the rotor is assigned one or more causes of unbalance with their respective unbalance distributions in the reference planes of the rotor, unbalances of each component are taken into account as a selectable quantile in an axial position, an assembly of a large number of rotors is simulated with its components using Monte Carlo simulation, the Monte Carlo simulation is recreated with a large number of rotors with their different distributions of causes of unbalance in the production process and from this the statistical unbalance distribution of the large number of rotors in their reference planes is derived. Procedure according to Claim 1 , characterized in that reference planes are introduced as compensation planes, so that the unbalance distribution must be designed as correctable by a balancing step. Method according to one of the preceding claims, characterized in that the causes of unbalance, unbalance values and axial positions of each component of the rotor are determined. Method according to one of the preceding claims, characterized in that imbalances of the components are related to bearing levels. Method according to one of the preceding claims, characterized in that the Monte Carlo simulation can also simulate a two-dimensional distribution of a systematic and a random unbalance deviation of all production steps. Method according to one of the preceding claims, characterized in that an intrinsic unbalance of each component is determined. Method according to one of the preceding claims, characterized in that each individual geometric tolerance limit which defines an imbalance amount is represented with an evenly distributed phase position over all rotors of the simulation.

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